Pseudo-ε expansion and renormalized coupling constants at criticality

Universal values of dimensional effective coupling constants g(2k) that determine nonlinear susceptibilities χ(2k) and enter the scaling equation of state are calculated for n-vector field theory within the pseudo-ε expansion approach. Pseudo-ε expansions for g(6) and g(8) at criticality are derived for arbitrary n. Analogous series for ratios R(6) = g(6)/g(4)(2) and R(8) = g(8)/g(4)(3) that figure in the equation of state are also found, and the pseudo-ε expansion for Wilson fixed point location g(4)(*) descending from the six-loop renormalization group (RG) expansion for the β function is reported. Numerical results are presented for 0 ≤ n ≤ 64, with the most attention paid to the physically important cases n = 0,1,2,3. Pseudo-ε expansions for quartic and sextic couplings have rapidly diminishing coefficients, so Padé resummation turns out to be sufficient to yield high-precision numerical estimates. Moreover, direct summation of these series with optimal truncation gives values of g(4)(*) and R(6)(*) that are almost as accurate as those provided by the Padé technique. Pseudo-ε expansion estimates for g(8)(*) and R(8)(*) are found to be much worse than those for lower-order couplings independently of the resummation method employed. The numerical effectiveness of the pseudo-ε expansion approach in two dimensions is also studied. Pseudo-ε expansion for g(4)(*) originating from the five-loop RG series for the β function of two-dimensional λϕ(4) field theory is used to get numerical estimates for n ranging from 0 to 64. The approach discussed gives accurate enough values of g(4)(*) down to n = 2 and leads to fair estimates for Ising and polymer (n = 0) models.